Date: Sun, 06 Jun 1999 22:38:37 -0700
Subject: word problem - four weights

Four Weights

When using a balance scale, weights can be placed on either side of the scale. For example, if a 10 pound weight provides a counter balance to an object and a 7 pound weight, then the object must weigh 3 pounds. What four weights can be used to weigh objects of 1, 2, 3 ... 38, 39, 40 pounds?

Hi,

There are a number of approaches running from trial and error to use of recurrences and the binomial theorem but the simplest way to look at this problem in general, i.e. to show how one can use n weights to balance the weights 1, 2, ... ,(3n - 1)/2, is to use what we call base 3 arithmetic. Without being too theoretical we will try to show you the algorithm of how to make a weight balance.
Recall that numbers that we commonly use such as 3456 are what we call base 10 numbers, i.e.

3456 = 3*103 + 4*102 + 5*101 + 6.

In other words
3456 = 3*1000 + 4*100 + 5*10 + 6.

This is a unique representation in terms of powers of 10. There is nothing special about powers of 10, we could just as easily write numbers in terms of powers of 3; for example, 70 is the same as
2*33 + 1*32 + 2*31 + 1 = 2*27 + 1*9 + 2*3 + 1.

We would say that the base 10 number 70 is 2121 in base 3, that is 70 = 21213. These 'digits' 2, 1, 2 and 1 are the coefficients of the different powers of 3 required. Similarly in base 3,
 2011023 = 2*35 + 0*34 + 1*33 + 1*32 + 0*31 + 2 = 2*243 + 1*27 + 1*9 +2 = 486 + 27 + 9 + 2 = 524.
Here are the first 40 base 10 numbers and their base 3 equivalents.

Base 10Base 3Base 10Base 3
11 21210
22 22211
310 23212
411 24220
512 25221
620 26222
721 271000
822 281001
9100 291002
10101 301010
11102 311011
12110 321012
13111 331020
14112 341021
15120 351022
16121 361100
17122 371101
18200 381102
19201 391110
20202 401111

Suppose that you want to balance say thirty-one pounds. Imagine that you have put thirty-one pounds on each side of the balance. On the left is thirty-one pounds written in base 10 notation (31) and on the right is twelve pounds written in base 3 notation (10113). Hence on the right is 1*33 + 1*31 + 1 so you can balance 31 pounds with one 27 pound weight, one 3 pound weight and one 1 pound weight. In a similar fashion any weight which has only ones and zeros when written in base 3 notation is easy to balance. For example, looking at the table above, 13 pounds balances with one 9 pound weight, one 3 pound weight and one 1 pound weight.
Suppose now that you want to weigh thirty-three pounds. Thirty-three in base 3 notation is 10203 but you don't have two 3 pound weights to put on the right. Put your 3 pound weight with the thirty-three pound weight on the left side of the balance and then to balance you need an extra three pounds on the right. That is on the right you need one 27 pound weight and three 3 pound weights which is equivalent to one 27 pound weight and one 9 pound weight. That is 33 + 3 balances with 10203 + 103 = 11003. Thus put your 1 pound weight on the left with the 33 pound weight and balance it with a 27 pound weight and a 9 pound weight.
As a second example look at seventeen pounds. Seventeen in base 3 is 1223. Since there is a 2 in the first position (reading from the right) put your 1 pound weight on the left, then to balance on the right you need eighteen pounds. Eighteen is 2003 which is two nines so again you need to put your 9 pound weight on the left, forcing another 9 pounds on the right for a sum of twenty-seven pounds. Thus 17 + 1 + 9 balances with 1223 + 13 + 1003 = 10003 which is 27 pounds. Thus to weight 17 pounds put the 1 and 9 pound weights on the left with the 17 pound weight and the 27 pound weight on the right.
What you are actually doing here is looking for a number which contains only zeros and ones when written in base 3, that when added to 1223 in base 3 results in a total that contains only zeros and ones. This number is 1013 since

1223
+1013
----
10003
Starting from the right, one plus two is three, but three is 103 so write 0 and carry the 1. In the next column two plus the one you carried is three, i.e. 103, so again write 0 and carry the 1. Finally one plus one plus one is three which is 103 so write 0 and carry the 1 for a total of 10003.

Cheers,
Penny

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